On the structure of separable $\mathcal {L} _\infty $-spaces

arXiv:1504.08223v1 [math.FA] 30 Apr 2015

ON THE STRUCTURE OF SEPARABLE L∞ -SPACES SPIROS A. ARGYROS, IOANNIS GASPARIS, AND PAVLOS MOTAKIS Abstract. Based on a construction method introduced by J. Bourgain and F. Delbaen, we give a general definition of a Bourgain-Delbaen space and prove that every infinite dimensional separable L∞ -space is isomorphic to such a space. Furthermore, we provide an example of a L∞ and asymptotic c0 space not containing c0 .

1. Introduction In the late 1960’s J. Lindenstrauss and A. Pelczy´ nski introduced the class of L∞ -spaces, which naturally extends the class of L∞ -spaces ([LP]). Whether such spaces always contain a copy of c0 remained a long standing open problem which was solved in the negative direction by J. Bourgain and F. Delbaen in [BD]. In particular, they introduced a method for constructing L∞ -spaces and any example constructed with its use has been customarily called a Bourgain-Delbaen space. This method proved to be very fruitful, as it has been used extensively to construct a wide variety of L∞ -spaces, including the first example of a Banach space satisfying the “scalar plus compact” property ([AH]), as well as to obtain other structural results in the geometry of Banach spaces ([FOS], [AFHORSZ]). The main framework of the Bourgain-Delbaen method concerns the construction of an increasing sequence (Yn )n of finite dimensional subspaces of a ℓ∞ (Γ) space, with Γ countably infinite, each one uniformly isomorphic to some ℓn∞ . This is achieved by carefully defining a sequence of extension operators (in )n , each one defined on ℓ∞ (Γn ) with Γn an appropriate finite subset of Γ, and taking Yn to be the image of in . If the sequence (in )n satisfies a certain compatibility property, then the closure of the union of the Yn , n ∈ N, is a L∞ -space whose properties depend on the definition of the aforementioned extension operators. Based on this method, we give a broad definition of a Bourgain-Delbaen space. We include a brief study of the basic properties of such spaces and using techniques rooted in the early theory of L∞ -spaces (i.e. [LP], [P], 2010 Mathematics Subject Classification: Primary 46B03, 46B06, 46B07 Key words: Bourgain-Delbaen spaces, separable L∞ -spaces, isomorphic ℓ1 -preduals, asymptotic c0 spaces. The authors would like to acknowledge the support of program APIΣTEIA-1082. 1

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S. A. ARGYROS, I. GASPARIS, AND P. MOTAKIS

[LR], [JRZ], [LS], [S]) we prove that every separable infinite dimensional L∞ -space is isomorphic to such a space. We use this result to deduce that every separable infinite dimensional L∞ -space X has an infinite dimensional L∞ -subspace Z, so that the quotient X/Z is isomorphic to c0 . In the final section of this paper we provide an example of an asymptotic c0 isomorphic ℓ1 -predual (i.e. a space whose dual is isomorphic to ℓ1 ) X0 that does not contain c0 . This result in particular yields that the proximity of a Banach space to c0 in a local setting, in the sense of being L∞ , as well as an asymptotic setting, in the sense of being asymptotic c0 , does not imply proximity to c0 in an infinite dimensional level. We think of this example as a step towards the solution of a problem in [GL, Question IV.2], namely whether every isomorphic ℓ1 -predual satisfying Pelczy´ nski’s property-(u) is isomorphic to c0 . The question of the existence of an asymptotic c0 L∞ space not containing c0 was asked by B. Sari. 2. Bourgain-Delbaen spaces In the present section we give a general definition of the spaces that shall be called Bourgain-Delbaen spaces and prove some of their basic properties. It turns out that every infinite dimensional separable L∞ -space is isomorphic to a Bourgain-Delbaen space. We recall the definition of a L∞ -space, which was introduced in [LP, Definition 3.1]. Definition 2.1. A Banach space X is called a L∞,C -space, for some C > 1, if for every finite dimensional subspace F of X there exists a finite dimensional subspace G of X, containing F , which is C-isomorphic to ℓn∞ , where n = dim G. A Banach space X will be called a L∞ -space, if it is a L∞,C space for some C > 1. Remark 2.2. If X is an infinite dimensional separable Banach space then it is well known, and not difficult to prove, that X is a L∞ -space if and only if there exists a constant C and an increasing sequence (Yn )n of finite dimensional subspaces of X, whose union is dense in X, such that Yn is C-isomorphic to ℓk∞n , where kn = dim Yn , for all n ∈ N. 2.1. The definition of a Bourgain-Delbaen space. We give a broad definition of what kind of spaces we will refer to as Bourgain-Delbaen spaces. Definition 2.3. Let Γ1 , Γ be non-empty sets with Γ1 ⊂ Γ. A linear operator i : ℓ∞ (Γ1 ) → ℓ∞ (Γ) will be called an extension operator, if for every x ∈ ℓ∞ (Γ1 ) and γ ∈ Γ1 we have that x(γ) = i(x)(γ). Definition 2.4. Let (Γq )∞ q=0 be a strictly increasing sequence of non-empty sets and Γ = ∪q Γq . A sequence of extension operators (iq )∞ q=0 , with iq : ℓ∞ (Γq ) → ℓ∞ (Γ) for all q ∈ N ∪ {0}, will be called compatible if for every p, q ∈ N ∪ {0} with p < q and x ∈ ℓ∞ (Γp ), the following holds: ip (x) = iq (rq (ip (x))),

ON THE STRUCTURE OF SEPARABLE L∞ -SPACES

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i.e. ip = iq ◦rq ◦ip , where rq : ℓ∞ (Γ) → ℓ∞ (Γq ) denotes the natural restriction operator. Definition 2.5. Let (Γq )∞ q=0 be a strictly increasing sequence of non-empty finite sets, Γ = ∪q Γq and (iq )∞ q=0 , with iq : ℓ∞ (Γq ) → ℓ∞ (Γ) for all q ∈ N ∪ {0}, be a sequence of compatible extension operators such that C = supq kiq k is finite. (i) We define the sets ∆0 = Γ0 and ∆q = Γq \ Γq−1 for q ∈ N ∪ {0}. (ii) For every γ ∈ Γ we define dγ , a vector in ℓ∞ (Γ), as follows: if γ ∈ ∆q for some q ∈ N ∪ {0}, then dγ = iq (eγ ). The space X(Γq ,iq )q = h{dγ : γ ∈ Γ}i, i.e. the closed subspace of ℓ∞ (Γ) spanned by the vectors (dγ )γ∈Γ , will be called a Bourgain-Delbaen space. Remark 2.6. Since the operators iq : ℓ∞ (Γq ) → ℓ∞ (Γ) are extension operators, it easily follows that iq is a C-isomorphism for all q ∈ N ∪ {0}, where C = supq kiq k. In particular, the following hold: (i) if Yq = iq [ℓ∞ (Γq )], then Yq is C-isomorphic to ℓ∞ (Γq ) and (ii) the vectors (dγ )γ∈∆q are C-equivalent to the unit vector basis of ℓ∞ (∆q ) for all q ∈ N ∪ {0}. Remark 2.7. The above Remark 2.6 and Proposition 2.12 from Subsection 2.2 provide an equivalent definition of a Bourgain-Delbaen space, namely the following. Let (Γq )q be a strictly increasing sequence of finite non-empty sets and (Yq )q be an increasing sequence of subspaces of ℓ∞ (Γ), where Γ = ∪q Γq . If there exists a constant C > 0 such that for every q ∈ N, when restricted onto the subspace Yq , the restriction operator rq : Yq → ℓ∞ (Γq ) is an onto C-isomorphism, then the space X = ∪q Yq is a Bourgain-Delbaen space. Indeed, it is straightforward to check that the maps iq : ℓ∞ (Γq ) → ℓ∞ (Γ) with iq = rq−1 : ℓ∞ (Γq ) → Yq ֒→ ℓ∞ (Γ) are uniformly bounded compatible extension operators and X(Γq ,iq )q = X. 2.2. Properties of a Bourgain-Delbaen space. We present some basic properties of Bourgain-Delbaen space, which can be deduced from Definition 2.5. Proposition 2.8. Let X(Γq ,iq )q be a Bourgain-Delbaen space. For all q ∈ N ∪ {0} we denote by Mq = h{dγ : γ ∈ ∆q }i. Then (Mq )∞ q=0 forms a finite dimensional decomposition (FDD) for the space X(Γq ,iq )q . More precisely, for every p, q ∈ N ∪ {0} with p < q and xℓ ∈ Mℓ for ℓ = 0, 1, . . . , q, the following holds:

q

p

X

X



xℓ (1) xℓ 6 C



ℓ=0

where C = supq kiq k.

ℓ=0

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S. A. ARGYROS, I. GASPARIS, AND P. MOTAKIS

Proof. Let xℓ = iℓ (yℓ ) for some yℓ ∈ h{eγ : γ ∈ ∆ℓ }i, for ℓ = 0, 1, . . . , q. Then, by the compatibility of the operators, we have that xℓ = ip (rp (iℓ (yℓ ))) i.e. xℓ = ip ◦ rp (xℓ ) for ℓ = 0, 1, . . . , p and hence:

p



! ! p p

X

X X



(2) xℓ = ip ◦ rp xℓ 6 C rp xℓ .



ℓ=0

ℓ=0

ℓ=0

On the other hand, once more by the extension property of the operators, we have that rp (xℓ ) = 0 (which also yields that ip ◦rp (xℓ ) = 0) for ℓ = p+1, . . . , q and therefore we obtain:



q ! ! p q



X X X





xℓ . xℓ = rp x ℓ > rp (3)





ℓ=0

ℓ=0

ℓ=0

The desired inequality immediately follows from (2) and (3).



Remark 2.9. By the proposition above, for every interval E = {p, . . . , q} of N ∪ {0} we can define the projection PE : X(Γq ,iq )q → Mp + · · · + Mq , associated to the FDD (Mq )q and the interval E. The above proof implies that (4)

P[0,q] x = iq ◦ rq (x)

for all q ∈ N ∪ {0} and x ∈ X(Γq ,iq )q and hence also (5)

P(p,q] x = iq ◦ rq (x) − ip ◦ rp (x)

for all p, q ∈ N ∪ {0} with p < q and x ∈ X(Γq ,iq )q . We shall call PE the Bourgain-Delbaen projection onto E. Note that kPE k 6 2C for every interval E of N ∪ {0}. Remark 2.10.
Proposition 2.8 in conjunction with Remark 2.6 (ii) yield ∞ that (dγ )γ∈∆q q=0 is a Schauder basis of X(Γq ,iq )q . Although in some cases it is more convenient to use the FDD (Mq )q , in Section 5 we shall indeed use this basis. Remark 2.11. Let x be a finitely supported vector in X(Γq ,iq )q with ran x = E. Using Remark 2.9 one can see that if q = max E, there exists y ∈ ℓ∞ (Γq ) with supp y ⊂ ∪p∈E ∆p such that x = iq (y). Proposition 2.12. Every Bourgain-Delbaen space X(Γq ,iq )q is a L∞ -space. More precisely, if Yq = iq [ℓ∞ (Γq )] for all q ∈ N ∪ {0}, then (Yq )q is a strictly increasing sequence of subspaces of X(Γq ,iq )q , whose union is dense in the whole space and for every q ∈ N ∪ {0} Yq is C-isomorphic to ℓ∞ (Γq ), where C = supq kiq k. Proof. By Remark 2.6 (i) we have that Yq is C-isomorphic to ℓ∞ (Γq ) for all q ∈ N ∪ {0}. It remains to show that the sequence (Yq )q is strictly increasing and its union is dense in X(Γq ,iq )q . We will prove that Yq = M0 + · · · + Mq , where Mq = h{dγ : γ ∈ ∆q }i for all q ∈ N ∪ {0}. Note that, in conjunction with Proposition 2.8, the previous fact easily implies the desired result.

ON THE STRUCTURE OF SEPARABLE L∞ -SPACES

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Note that for p, q ∈ N ∪ {0} with p 6 q and x ∈ Mp there is a y ∈ h{eγ : γ ∈ ∆p }i with x = ip (y). Using the compatibility of the extension operators we obtain that x = iq (rq (x)) ∈ iq (ℓ∞ (Γq )) = Yq . We have hence concluded that M0 + · · · + Mq ⊂ Yq . To conclude that the above inclusion cannot be proper, we will show that dim(M0 + · · · + Mq ) = dim Yq . Note that since, Proposition 2.8, (Mq )∞ q=0 is an FDD, we have that dim(M0 +· · ·+ Mq ) = dim(M0 ) + · · · + dim(Mq ). Moreover, Remark 2.6 (ii) implies that dim(Mp ) = #∆p for p = 0, . . . , q. The definition of the sets ∆p yields that dim = (M0 + · · · + Mq ) = #Γq . Remark 2.6 (i) implies that dim Yq = #Γq and therefore dim(M0 + · · · + Mq ) = dim Yq .  2.3. The functionals (e∗γ )γ∈Γ . Let X(Γq ,iq )q be a Bourgain-Delbaen space. For every γ ∈ Γ we denote by e∗γ : X(Γq ,iq )q → R the evaluation functional on the γ’th coordinate, defined on ℓ∞ (Γ) and then restricted to the subspace X(Γq ,iq )q . Note that ke∗γ k 6 1 for all γ ∈ Γ. Proposition 2.13. Let X(Γq ,iq )q be a Bourgain-Delbaen space. Then (e∗γ )γ∈Γ is C-equivalent to the unit vector basis of ℓ1 (Γ), where C = supq kiq k. Proof. Let A be a non-empty finite subset of Γ and (λγ )γ∈A be scalars. Choose P q ∈ N ∪ {0} such that A ⊂ Γq and take the normalized vector y = γ∈A sgn λγ eγ in ℓ∞ (Γq ). Note that kiq (y)k 6 C. Since iq is an extension operator and A ⊂ Γq we obtain the following estimate:

X

X X X

∗

λγ eγ > λγ e∗γ (iq (y)) = λγ iq (y)(γ) C |λγ | > C

γ∈A

γ∈A γ∈A γ∈A X X X = λγ y(γ) = λγ sgn λγ = |λγ |. γ∈A

i.e. C

P −1

|λγ | 6 k

P

λγ e∗γ k 6

γ∈A

P

γ∈A

|λγ |, which is the desired result.



Definition 2.14. Let X(Γq ,iq )q be a Bourgain-Delbaen space. For every γ ∈ Γ we define two bounded linear functionals c∗γ , d∗γ : X(Γq ,iq )q → R as follows: (i) if γ ∈ ∆0 then c∗γ = 0 and otherwise if γ ∈ ∆q+1 for some q ∈ N∪{0}, then c∗γ = e∗γ ◦ iq ◦ rq and (ii) d∗γ = e∗γ − c∗γ for all γ ∈ Γ. Remark 2.15. By Remark 2.9 obtain that if q ∈ N∪{0} and γ ∈ ∆q+1 , then c∗γ = e∗γ ◦P[0,q] and hence d∗γ = e∗γ ◦P(q,∞) . Also, using the extension property of the operators iq , it easily follows from Remark 2.9 that for p ∈ N ∪ {0} and γ ∈ Γp we have that e∗γ = e∗γ ◦ P[0,p] which also implies that if γ ∈ ∆p , then d∗γ = e∗γ ◦ P{p} . Moreover, for γ ∈ ∆0 , d∗γ = e∗γ . Lemma 2.16. Let X(Γq ,iq )q be a Bourgain-Delbaen space. For all q ∈ N∪{0} and γ0 ∈ ∆q+1 the functional c∗γ0 is in the linear span of {e∗γ : γ ∈ Γq }.

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Proof. It suffices to show that ∩γ∈Γq ker e∗γ ⊂ ker c∗γ0 and to that end let x ∈ X(Γq ,iq )q with e∗γ (x) = 0 for all γ ∈ Γq . Recall that x is also a vector in ℓ∞ (Γ) and in particular we have that x(γ) = e∗γ (x) = 0 for all γ ∈ Γq and hence, rq (x) = 0. By the definition of c∗γ0 it easily follows that c∗γ0 (x) = 0.  Proposition 2.17. Let X(Γq ,iq )q be a Bourgain-Delbaen space. The following hold. (i) The functionals (d∗γ )γ∈Γ are biorthogonal to the vectors (dγ )γ∈Γ . (ii) For all q ∈ N ∪ {0} we have that h{d∗γ : γ ∈ Γq }i = h{e∗γ : γ ∈ Γq }i. In particular, the closed linear span of the functionals (d∗γ )γ∈Γ is C-isomorphic to ℓ1 , where C = supq kiq k. (iii) If moreover the FDD (Mq )q is shrinking, then the closed linear span of the functionals (d∗γ )γ∈Γ is X∗(Γq ,iq )q . In particular, X∗(Γq ,iq )q is Cisomorphic to ℓ1 . Proof. The first statement easily follows from Remark 2.15, in particular the fact that for q ∈ N ∪ {0} and γ ∈ ∆q we have that d∗γ = e∗γ ◦ P{q} . For the proof of (ii) observe that by Lemma 2.16 and d∗γ = e∗γ − c∗γ , we have that h{d∗γ : γ ∈ Γq }i ⊂ h{e∗γ : γ ∈ Γq }i. Moreover (i) implies that dimh{d∗γ : γ ∈ Γq }i = #Γq and hence the inclusion cannot be proper. The last part of statement (ii) follows from Proposition 2.13. To prove the
last statement note that if (Mq )q is shrinking, then so is the ∞  basis (dγ )γ∈∆q q=0 . From (ii) the desired result follows. 3. Separable L∞ -spaces are Bourgain-Delbaen spaces

We combine some simple remarks concerning Bourgain-Delbaen spaces with results from [JRZ], [LS] and [S] to prove that whenever X is an infinite dimensional separable L∞ -space, then X is isomorphic to a BourgainDelbaen space. Lemma 3.1. Let (µi )i be a bounded sequence in M[0, 1]. Then there exists a sequence (ti )i of elements of [0, 1] such that if νi = µi + δti for all i ∈ N then the following hold: (i) the sequence (νi )i is equivalent to the unit vector basis of ℓ1 (N) and (ii) the space Y = [(νi )i ] is complemented in M[0, 1]. Proof. Find a sequence (ti )i of elements of [0, 1] so that µj ({ti }) = 0 for all i, j ∈ N. Setting νi = µi + δti , it can P be shown that (νi )i is equivalent to the basis of ℓ1 and that the map Sµ = ∞ i=1 µ({ti })νi is a bounded projection onto the space [(νi )i ].  Lemma 3.2. Let X be an infinite dimensional and separable L∞ -space. Then there exists a sequence (x∗i )i in X ∗ satisfying the following: (i) the sequence (x∗i )i is equivalent to the unit vector basis of ℓ1 (N), (ii) there exists a constant θ > 0 such that θkxk 6 supi |x∗i (x)| for all x ∈ X and

ON THE STRUCTURE OF SEPARABLE L∞ -SPACES

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(iii) the space Y = [(x∗i )i ] is complemented in X ∗ . Proof. As it is shown in [S], the dual of X is either isomorphic to ℓ1 (N) or to M[0, 1] (see also [LS]). In the first case, just choose a Schauder basis (x∗i )i of X ∗ which is equivalent to the unit vector basis of ℓ1 (N). In the second case, let T : X ∗ → M[0, 1] be an onto isomorphism and choose a normalized sequence (zi∗ )i in X ∗ such that kxk = supi |zi∗ (x)| for all x ∈ X. Fix C > kT −1 k and apply Lemma 3.1 to the sequence (µi )i with µi = CT zi∗ for all i ∈ N to find a sequence (ti )i in [0, 1] such that if νi = µi + δti for all i ∈ N then (νi )i satisfies the conclusion of that lemma. Setting x∗i = T −1 νi for all i ∈ N, it is not hard to check that, for θ = C − kT −1 k, the sequence (x∗i )i is the desired one.  Lemma 3.3. Let X be a L∞,λ -space, Y be a subspace of X ∗ and assume that there exists a constant θ > 0 so that for all x ∈ X, θkxk 6 sup{|y ∗ (x)| : y ∗ ∈ BY ∗ }. Then for every finite dimensional subspace F of X and every ε > 0 there exists a finite rank operator T : X → X satisfying the following: (i) kT x − xk 6 εkxk for all x ∈ F , (ii) kT k 6 λ/θ, (iii) T ∗ [X ∗ ] ⊂ Y . In particular, if X is separable then there exits a sequence of finite rank operators Tn : X → X with Tn x → x for all x ∈ X and Tn∗ [X ∗ ] ⊂ Y for all n ∈ N. Proof. Let F be a finite dimensional subspace of X and ε > 0. By passing to a larger subspace, we may assume that F = h{yi : i = 1, . . . , n}i, where the map A : ℓn∞ → F with Aei = yi is invertible with kAkkA−1 k 6 λ. By the Hahn-Banach theorem there is a sequence (yi∗ )ni=1 in X ∗ , biorthogonal to (yi )ni=1 , such that kyi∗ k 6 kA−1 k for i = 1, . . . , n. A separation theorem
w∗ and hence, we may choose y˜i∗ ∈ θ −1 kA−1 k BY yields that BX ∗ ⊂ θ −1BY such that yi∗ − yi∗ )|F k < ε/kAk for i = 1, . . . , n. Define T : X → X with Pn k(˜ T x = i=1 y˜i∗ (x)yi . Some standard calculations yield that T is the desired operator.  The following terminology is from [JRZ]. If E1 , E2 are subspaces of a Banach space X and ε > 0, we say that E2 is ε-close to E1 , if there is an invertible operator T from E1 onto E2 with kT x − xk 6 εkxk for all x ∈ E1 . Note that if E2 is ε-close to E1 , then E1 is ε/(1 − ε)-close to E2 . ∗ Lemma 3.4. Let X be a Banach space and (x∗i )∞ i=1 be a sequence in X which is equivalent to the unit vector basis of ℓ1 (N). Assume moreover that there exists a sequence of bounded finite rank projections (Qn )n satisfying the following:

(i) Qn : X → X, Qn [X] ⊂ Qn+1 [X] for all n ∈ N and Qn x → x for all x ∈ X.

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S. A. ARGYROS, I. GASPARIS, AND P. MOTAKIS

(ii) There is a strictly increasing sequence of natural numbers (kn )n and 0 < ε < 1 with ε/(1−ε) < (supn kQn k)−1 such that Q∗n [X ∗ ] is ε-close to the space h{x∗i : i = 1, . . . , kn }i for all n ∈ N. Then X is isomorphic to a Bourgain-Delbaen space. Proof. Note that (i) implies that ∪n Qn [X] is dense in X. Define a linear operator U : X → ℓ∞ (N) with U x = (x∗i (x))i for all x ∈ X, evidently U is bounded. Set Γn = {1, . . . , kn } for all n ∈ N and Yn = U Qn [X]. We shall prove that U is an isomorphic embedding and that the restriction operators onto the first kn coordinates rkn : Yn → ℓ∞ ({1, . . . , kn }) are onto C-isomorphisms for all n ∈ N, for a uniform constant C. Given the aforementioned facts and Remark 2.7, the desired result follows easily. Since ∪n Qn [X] is dense in X, to show that U is an isomorphic embedding, it is enough to find a uniform constant c > 0, such that kU xk > ckxk for every x ∈ Qn [X] and for every n ∈ N. To this end let S : [(x∗i )i ] → ℓ1 (N) be the onto isomorphism with Sx∗i = ei for all i ∈ N. Let n ∈ N and x ∈ Qn [X]. The Hahn-Banach Theorem and the fact that h{x∗i : i = 1, . . . , kn }i is ε/(1 − ε)-close to Q∗n [X ∗ ], in conjunction with some tedious computations, yield that there exists an i0 ∈ {1, . . . , kn } such that (6)

|x∗i0 (x)| > (1 − ε)

ε 1 − 1−ε kQn k kxk. kSkkQn k

Setting c = (1 − ε)(1 − (ε/(1 − ε)) supk kQk k)/(kSk supk kQk k) we conclude kU xk > ckxk. It remains to show that there a exists a constant C > 0 such that rkn : Yn → ℓ∞ ({1, . . . , kn }) is an onto C-isomorphism for every n ∈ N. Let n ∈ N and z ∈ Yn . Set x = U −1 z ∈ Qn [X]. Then by (6) there is an i0 ∈ {1, . . . , kn } such that: c kzk > krkn (z)k > |x∗i0 (x)| > ckxk > kzk. kU k Setting C = kU k/c we conclude that rkn |Yn is a C-isomorphic embedding. Finally, observe that dim Yn = dim Qn [X] = dim Q∗n [X ∗ ] = kn and hence rkn |Yn is also onto.  The following lemma and its proof can be found in [JRZ, Lemma 4.3]. Lemma 3.5. Let X be a separable Banach space and Y be a separable subspace of X ∗ . Assume moreover that (Pn )n , (Tn )n are bounded finite rank operators satisfying the following conditions: (a) Pn : X → X, Pn∗ [X ∗ ] ⊂ Y and Tn : X ∗ → Y for all n ∈ N. (b) Pn x → x for all x ∈ X, Tn y → y for all y ∈ Y and supn kTn k < ∞. (c) The operators (Tn )n are projections. If E, F are finite dimensional subspaces of X, X ∗ respectively and 0 < ε < 1, then there exists a projection Q : X → X with finite dimensional range such that: (i) Qe = e for all e ∈ E,

ON THE STRUCTURE OF SEPARABLE L∞ -SPACES

(ii) (iii) (iv) (v)

9

Q∗ f = f for all f ∈ F , Q∗ [X ∗ ] ⊂ Y , kQk 6 2c + 2K + 4cK where c = supn kPn k and K = supn kTn k, Q∗ [X ∗ ] is ε-close to Tn [X ∗ ] for some integer n.

Theorem 3.6. Every separable infinite dimensional L∞ -space is isomorphic to a Bourgain-Delbaen space. Proof. Let X be a separable infinite dimensional L∞ -space. By Lemma 3.2, there exists a sequence (x∗i )i in X ∗ which is equivalent to the unit vector basis of ℓ1 (N), a constant θ > 0 such that θkxk 6 supi |x∗i (x)| for all x ∈ X and a bounded linear projection P : X ∗ → Y = [(x∗i )i ]. For n ∈ N define Tn : X ∗ → Y with Tn = Sn ◦ P , where Sn : Y → Y denotes the basis projection of (x∗i )i onto the first n coordinates. Also apply Lemma 3.3 to find a sequence of finite rank operators Pn : X → X such that Pn x → x for all x ∈ X and Pn∗ [X ∗ ] ⊂ Y for all n ∈ N. Assumptions (a), (b) and (c) of Lemma 3.5 are evidently satisfied. Choose ε > 0 with ε/(1 − ε) < 1/(2c + 2K + 4cK) where c = supn kPn k and K = supn kTn k. Recursively, setting En = Pn [X] + Qn−1 [X], where Q0 is the zero operator on X, and Fn = {0}, using Lemma 3.5, choose (Qn )n a sequence of bounded linear projections on X such that Pn [X] ⊂ Qn [X] ⊂ Qn+1 [X], kQn k 6 2c + 2K + 4cK for all n ∈ N and there exist natural numbers (kn )n , such that Q∗n [X ∗ ] is ε-close to Tkn [X ∗ ] = h{x∗i : i = 1, . . . , kn }i for all n ∈ N. Note that dim Qn [X] → ∞ and hence, by passing to a subsequence we may assume that (kn )n is strictly increasing. We conclude that the sequences (x∗i )∞ i=1 and (Qn )n witness the fact that the space X satisfies the assumptions of Lemma 3.4 and therefore it is isomorphic to a Bourgain-Delbaen space.  4. Separable infinite dimensional L∞ -spaces contain L∞ -subspaces of infinite co-dimension Using the main result of Section 3 we prove that every infinite dimensional L∞ -space X contains an infinite dimensional L∞ -subspace Z so that the quotient X/Z is isomorphic to c0 . Lemma 4.1. Let X(Γq ,iq )q be a Bourgain-Delbaen space and assume that there exists a decreasing sequence of positive real numbers (εq )∞ q=1 , with P 2C 2 q εq < 1 where C = supq kiq k, so that for every q ∈ N there exist γ1q , γ2q ∈ ∆q with γ1q 6= γ2q satisfying ke∗γ q ◦ ip ◦ rp − e∗γ q ◦ ip ◦ rp k < εq for 1 2 p = 0, 1, . . . , q − 1. Then X(Γq ,iq )q contains an infinite dimensional L∞ subspace Z. Proof. Define Rq = {γ1q , γ2q }, Sq = ∪qp=1 Rp for q ∈ N and S = ∪∞ q=1 Sq . Define N0 = M0 = h{dγ : γ ∈ ∆0 }i and Nq = h{dγ : γ ∈ ∆q \ Rq } ∪ {dγ1q + dγ2q }i

10

S. A. ARGYROS, I. GASPARIS, AND P. MOTAKIS

for q ∈ N. Note that Nq is a subspace of Mq of co-dimension one for every q ∈ N. Define Zq = h{dγ : γ ∈ Γq \ Sq } ∪ {dγ1p + dγ2p : p = 1, . . . , q}i for all q ∈ N. Observe that Zq = N0 + · · · + Nq for all q ∈ N and that (Zq )q is an increasing sequence of finite dimensional spaces whose union is dense in the space Z = h{dγ : γ ∈ Γ \ S} ∪ {dγ1q + dγ2q : q ∈ N}i. We shall prove that Z is the desired subspace. To begin, Z is clearly infinite dimensional. More precisely, the spaces (Nq )q form an FDD for the space Z with a projection constant C. To conclude that Z is indeed a L∞ -space, it suffices to show that P∞ for each nq 2 q ∈ N the space Zq is 1+ε C-isomorphic to ℓ where ε = 2C ∞ q=1 εq and 1−ε nq = dim Zq = #Γq −q. For q ∈ N we define a subspace of ℓ∞ (Γq ) as follows: Wq = h{eγ : γ ∈ Γq \ Sq } ∪ {eγ1p + eγ2p : p = 1, . . . , q}i. n

fq = iq [Wq ]. Observe that Wq is isometric to ℓ∞q and therefore, Also define W fq is C-isomorphic to ℓn∞q . Hence, if we find a linear map by Remark 2.6, W fq with kTq x − xk 6 εkxk for all x ∈ Zq the proof will be Tq : Zq → W complete. Let us fix q ∈ N and observe that if x ∈ Np , for some 0 6 p 6 q, then x = ip (y) where y ∈ h{eγ : γ ∈ ∆q \ Sq } ∪ {eγ1q + eγ2q }i. For p = 0, . . . , q we define Tq,p : Np → Wq as follows:  y(γ) if rank(γ) 6 p,    ip (y)(γ) if rank(γ) > p and γ ∈ / Sq , Tq,p (x)(γ) = s ip (y)(γ1 ) if rank(γ) > p and γ = γ1s for some s ∈ (p, q],    ip (y)(γ1s ) if rank(γ) > p and γ = γ2s for some s ∈ (p, q].

Observe that the map is linear and well defined. Also observe that if x(γ) 6= Tq,p (x)(γ), for some γ ∈ Γq , then necessarily there is an s ∈ (p, q] so that γ = γ2s and Tq,p (x)(γ) = ip (y)(γ1s ) = e∗γ s ◦ip ◦rp (x). Hence, for such a γ ∈ Γq : 1

|x(γ) − Tq,p (x)(γ)| =

|e∗γ2s

◦ ip ◦ rp (x) − e∗γ1s ◦ ip ◦ rp (x)| < εs kxk 6 εp+1 kxk.

We conclude that krq (x) − Tq,p (x)k 6 εp+1 kxk (actually observe that if fq with p = q then rq (x) = Tq,q (x)) and hence if we define Teq,p : Np → W Teq,p = iq ◦ Tq,p then



x − Teq,p (x) 6 Cεp+1 kxk

for all x ∈ Np and p = 0, . . . , q. As we previously mentioned, (Np )p is an FDD for Z with a projection constant C, so we may consider the associated fq with Tq = projections Q{p} : Z → Np for all p. Define Tq : Zq → W Pq e p=0 Tq,p ◦ Q{p} . Some simple calculations using kQ{p} k 6 2C for all p yield that Tq is the desired operator. 

ON THE STRUCTURE OF SEPARABLE L∞ -SPACES

11

We recall that it is known that separable L∞ -spaces have c0 as a quotient [P], [AB]. Lemma 4.2. Let X(Γq ,iq )q be a Bourgain-Delbaen space satisfying the assumptions of Lemma 4.1. If Z is the subspace constructed in that lemma then the quotient X(Γq ,iq )q /Z is isomorphic to c0 (N). Proof. Following the notation of the proof of Lemma 4.1, let Z be the constructed subspace and set also W be the closed linear span of the vectors dγ1q − dγ2q , q ∈ N. Let Q : X(Γq ,iq )q → X(Γq ,iq )q /Z denote the corresponding quotient map and for q ∈ N define yq = Q(dγ1q − dγ2q ). It is not very difficult to prove that (yq )q is a Schauder basis of X(Γq ,iq )q /Z with projection constant C = supq kiq k. Let wq∗ = 1/2(d∗γ q − d∗γ q ) for all q ∈ N and 1 2 (yq∗ )q ⊂ (X(Γq ,iq )q /Z)∗ be the biorthogonal sequence of (yq )q . It follows that the sequences (wq∗ )q and (yq∗ )q are isometrically equivalent and that kwq∗ − 2(e∗γ q − e∗γ q )k 6 2εq for all q. Proposition 2.13 yields that (yq∗ )q is 1 2 equivalent to the unit vector basis of ℓ1 , which indeed implies that (yq )q is equivalent to the unit vector basis of c0 (N).  Proposition 4.3. Every separable infinite dimensional L∞ -space X contains an infinite dimensional L∞ -subspace Z so that the quotient X/Z is isomorphic to c0 (N). In other words, every separable infinite dimensional L∞ -space X is the twisted sum of a L∞ -space Z and c0 (N). Proof. By virtue of Theorem 3.6, we may assume that X is a BourgainDelbaen space X(Γq ,iq )q . We start by observing that for every strictly increasing sequence (qs )∞ s=0 of N ∪ {0}, the Bourgain-Delbaen spaces X(Γq ,iq )q and X(Γqs ,iqs )s are actually equal. This follows from Proposition 2.12, in particular the fact that ∪q Yq is dense in X(Γq ,iq )q . It is therefore sufficient to find an appropriate increasing sequence (qs )∞ s=0 so that the space X(Γqs ,iqs )s satisfies the assumptions of Lemma 4.1 (and hence also those of Lemma P 4.2). , with Fix a decreasing sequence of positive real numbers (εq )∞ q=1 q εq < 2 1/(2C ) where C = supq kiq k. A compactness argument yields that for every ε > 0 and q ∈ N ∪ {0} there exist non-equal γ1 , γ2 ∈ Γ \ Γq so that ke∗γ1 ◦ip ◦rp −e∗γ2 ◦ip ◦rp k < ε for p = 0, . . . , q. Set q0 = 0 and using the above, recursively choose a strictly increasing sequence (qs )∞ s=0 so that for every s ∈ N there are non-equal γ1s , γ2s ∈ Γqs \Γqs−1 with ke∗γ 1 ◦iqt ◦rqt −e∗γ 2 ◦iqt ◦rqt k 6 εs s s for t = 0, . . . , s − 1.  5. A L∞ and asymptotic c0 space not containing c0 We employ the Bourgain-Delbaen method to define an isomorphic ℓ1 predual X0 , which is asymptotic c0 and does not contain a copy of c0 . We follow notation similar to that used in [BD] and [AH]. 5.1. Definition of the space X0 . We fix a natural number N > 3 and a constant 1 < θ < N/2. Define ∆0 = {0} and assume that we have defined

12

S. A. ARGYROS, I. GASPARIS, AND P. MOTAKIS

the sets ∆0 , ∆1 , . . . , ∆q . We set Γp = ∪pi=0 Γi and for each γ ∈ Γq We denote by rank(γ) the unique p so that γ ∈ ∆p . Assume that to each γ ∈ Γq \ Γ0 we have assigned a natural number in {1, . . . , n}, called the age of γ and denoted by age(γ). Assume moreover that we have defined extension functionals (c∗γ )γ∈∆p and extension operators ip−1,p : ℓ∞ (Γp−1 ) → ℓ∞ (Γp ) so that  x(γ) if γ ∈ Γp−1 , (7) ip−1,p (x)(γ) = c∗γ (x) if γ ∈ ∆p for p = 1, . . . , q. If 0 6 p < s 6 q we define ip,s = is−1,s ◦ · · · ◦ ip,p+1 and we also denote by ip,p the identity operator on ℓ∞ (Γp ). We define ∆q+1 to be the set of all tuples of one of the two forms described below:   (8a) q + 1, n, (εi )ki=1 , (Ei )ki=1 , (ηi )ki=1

where n 6 (#Γq )2 , 1 6 k 6 n, εi ∈ {−1, 1} for i = 1, . . . , k, (Ei )ki=1 is a sequence of successive intervals of {0, . . . , q} and ηi ∈ Γq with rank(ηi ) ∈ Ei for i = 1, . . . , k, or   (8b) q + 1, ξ, n, (εi )ki=1 , (Ei )ki=1 , (ηi )ki=1

where ξ ∈ Γq−1 \Γ0 with age(ξ) < N , (#Γrank(ξ) )2 6 n 6 (#Γq )2 , 1 6 k 6 n, εi ∈ {−1, 1} for i = 1, . . . , k, (Ei )ki=1 is a sequence of successive intervals of {rank(ξ) + 1, . . . , q} and ηi ∈ Γq with rank(ηi ) ∈ Ei for i = 1, . . . , k. To each γ ∈ ∆q+1 we assign an extension functional c∗γ : ℓ∞ (∆q ) → R. If q x = ip,q ◦rp (x) while if 0 6 p 6 s 6 q 0 6 p 6 q we define the projection P[0,p] q q q . If γ ∈ ∆q+1 is of the form we define the projection P(p,s] = P[0,s] − P[0,p] (8a), we set age(γ) = 1 and define the extension functional c∗γ as follows: k

(9a)

c∗γ

θ 1X ∗ εi eηi ◦ PEq i . = Nn i=1

If γ ∈ ∆q+1 is of the form (8b), we set age(γ) = age(ξ) + 1 and define the extension functional c∗γ : ℓ∞ (∆q ) → R as follows: k

(9b)

c∗γ = e∗ξ +

θ 1X ∗ εi eηi ◦ PEq i . Nn i=1

The inductive construction is complete. We set Γ = ∪∞ q=1 ∆q and for each q ∈ N ∪ {0} we define the extension operator iq : ℓ∞ (Γq ) → ℓ∞ (Γ) by the rule  x(γ) if γ ∈ Γq , iq (x)(γ) = iq,p (x)(γ) if γ ∈ ∆p for some p > q.

Standard arguments yield that (iq )∞ q=0 is a compatible sequence of extension operators with supq kiq k 6 N/(N − 2θ). We denote by X0 the resulting

ON THE STRUCTURE OF SEPARABLE L∞ -SPACES

13

Bourgain-Delbaen space X(Γq ,iq )q . We shall use the notation from section 2.1. Remark 5.1. By Remark 2.9 we obtain that for every interval E of the natural numbers, kPE k 6 2N/(N − 2θ). It follows that for every γ ∈ Γ and such E, ke∗γ ◦ PE k 6 2N/(N − 2θ), in particular kd∗γ k 6 2N/(N − 2θ). Remark 5.2. We enumerate the set Γ = ∪∞ q=0 ∆q in such a manner that the sets ∆q correspond to successive intervals of N. If we denote this enumeration by Γ = {γi : ∈ N}, according to Remark 2.10, (dγi )i forms a Schauder basis for X0 . It is with respect to this basis that we show that the space X0 is asymptotic c0 . However, when we write PE we shall mean the Bourgain-Delbaen projection onto E as defined in Remark 2.9. Arguing as in [AH, Proposition 4.5], each e∗γ admits an analysis. Proposition 5.3. Let γ ∈ Γ with rank(γ) > 0. The functional e∗γ admits a unique analysis of the following form: (10)

e∗γ =

a X r=1

d∗ξr +

kr a θ X 1 X εr,i e∗ηr,i ◦ PEr,i N nr r=1

i=1

where a = age(γ) 6 N , the ξi ’s are in Γ \ Γ0 with ξa = γ, the Er,i ’s are intervals of N ∪ {0} with E1,1 < · · · < E1,k1 < rank(ξ1 ) < E2,1 < · · · < Ea,ka < rank(ξa ), the ηr,i ’s are in Γ with rank(ηr,i ) ∈ Er,i , the εr,i ’s are in {−1, 1}, nr > (#Γrank(ξr−1 ) )2 for r = 2, . . . , a and 1 6 kr 6 nr for r = 1, . . . a. P r ∗ Remark 5.4. Note that if in (10) we set fr = 1/nr ki=1 eηr,i ◦ PEr,i , then (fr )ar=1 constitutes a very fast growing sequence of α-averages, in the sense of [ABM]. 5.2. The main property of the space X0 . Lemma 5.5. Let x1 < · · · < xm be blocks of (dγi )i of norm at most one, E1 < · · · < Ek be intervals of N ∪ {0}, (ηi )ki=1 be a sequence in Γ with rank(ηi ) ∈ Ei , for i = 1, . . . , k, (εi )ki=1 be a sequence in {−1, 1} and let n ≥ max{m2 , k}. Then  ! m k X 1X 4N ∗ xj  6 εi eηi ◦ PEi  . (11) n N − 2θ j=1 i=1

Proof. We shall consider the support and range of vectors with respect to the basis (dγi )i . Set x∗i = εi e∗ηi ◦ PEi for i = 1, . . . , k. Note that (x∗i )ki=1 is a successive block sequence of (d∗γi )i and by Remark 5.1, kx∗i k ≤ 2N/(N − 2θ) for i = 1, . . . , k. Let I1 be the set of those i ≤ k such that the support of x∗i intersects thePrange of at least two of the xj ’s. Then #I1 ≤ m and so P |(1/n i∈I1 x∗i )( m 2θ))m2 /n ≤ 2N/(N − θ). Let I2 = j=1 xj )| 6 (2N/(N −P P {i ≤ n : i ∈ / I1 }. It is clear that |(1/n i∈I2 x∗i )( m j=1 xj )| 6 2N/(N − 2θ) and the proof is complete. 

14

S. A. ARGYROS, I. GASPARIS, AND P. MOTAKIS

Proposition 5.6. Set KN,θ = (2N 3 + 4θN 2 − 4θN )/(N 2 − 3θN + 2θ 2 ) and let u1 < · · · < um be blocks of (dγi )i of norm at most equal to one so that if we consider the support of the ui with respect to the basis (dγi )i Pvectors m then, m ≤ min supp u1 . Then k i=1 ui k 6 KN,θ . P Proof. Set u = m i=1 ui . We use induction on rank(γ) to show that for every γ ∈ Γ and every interval E of N, |e∗γ ◦ PE (u)| 6 KN,θ . This assertion is easy when rank(γ) = 0. Assume that q ∈ N ∪ {0} is such that the assertion holds for every γ ∈ Γq and E interval of N and let γ ∈ Γq+1 \ Γq and E be an interval of N. Applying Proposition 5.3, write e∗γ =

a X

d∗ξr +

kr a θ X 1 X εr,i e∗ηr,i ◦ PEr,i N nr r=1

r=1

i=1

so that the conclusion of that proposition is satisfied. For r = 1, . . . , a define P r ε e∗ ◦ PEr,i ∩E and G = {r : rank(ξr ) ∈ E}. Observe yr∗ = (1/nr ) ki=1 P P r,i η∗r,i ∗ that eγ ◦ PE = r∈G dξr + (θ/N ) ar=1 yr∗ . If ξr = γir note that i1 < · · · < ia . Considering the support of u with respect to the basis (dγi )i , set r0 = min{r : ir > min supp u}. The inductive assumption yields that |yr∗0 (u)| 6 KN,θ , while the growth condition on the nr ’s implies that nr > m2 for r > r0 . Lemma 5.5 yields   a θ X ∗ θ 4N (12) |y (u)| 6 KN,θ + (N − 1) N r=1 r N N − 2θ while Remark 5.1 implies (13)

X

r∈G

|d∗ξr (u)| 6 N

2N . N − 2θ

Some calculations combining (12) and (13) yield the desired result.



Proposition 5.7. The space X0 is a L∞ -space with a Schauder basis (dγi )i satisfying the following properties. (i) The basis (dγi )i is shrinking, in particular X∗0 is isomorphic to ℓ1 . (ii) The space X0 is asymptotic c0 with respect to the basis (dγi )i . (iii) The space X0 does not contain an isomorphic copy of c0 . Proof. The fact that X0 is asymptotic c0 with respect to (dγi )i follows directly from Proposition 5.6. We obtain in particular that (dγi )i is shrinking and hence, by Proposition 2.17, the dual of X0 is isomorphic to ℓ1 . To show that X0 does not contain an isomorphic copy of c0 , let us consider the FDD (Mq )∞ q=0 as it was defined in Proposition 2.8. Let (xk )k be a sequence of skipped block vectors, with respect to the FDD (Mq )∞ q=0 , all of which have norm at least equal to one. Then for every ε > 0 there exist a finite subset P I1 of N and γ ∈ Γ so that e∗γ ( k∈I1 xk ) > θ − ε. It follows from this that for P all n ∈ N we can find a γ ∈ Γ and find a finite subset J of N so that e∗γ ( i∈J xk ) > (θ −ε)n . Therefore, c0 is indeed not isomorphic to a subspace of X0 . 

ON THE STRUCTURE OF SEPARABLE L∞ -SPACES

15

Remark 5.8. In [A] D. E. Alspach proved that the L∞ -space with separable dual defined in [BD] has Szlenk index ω. We note that the space X0 has Szlenk index ω as well. Indeed, if this were not the case then by [AB, Theorem 1] we would conclude that X0 has a quotient isomorphic to C(ω ω ) which would imply that X0 admits an ℓ1 spreading model. Remark 5.9. Every skipped block sequence, with respect to the FDD (Mq )∞ q=1 , is boundedly complete and hence, the space X0 is reflexively saturated and also every block sequence in X0 contains a further block sequence which is unconditional. We also note that a similar method can be used to construct a reflexive asymptotic c0 space with an unconditional basis. This space is related to Tsirelson’s original Banach space [T]. References [A] D. E. Alspach, The dual of the Bourgain-Delbaen space, Israel J. Math. bf 117 (2000), 239-259. [AB] D. E. Alspach and Y. Benyamini, C(K) quotients of separable L∞ spaces, Israel J. Math. 32 (1979), 145-160. [ABM] S. A. Argyros, K. Beanland and P. Motakis, The strictly singular operators in Tsirelson like reflexive spaces, Illinois J. Math. 57 (2013), no. 4, 1173-1217. [AFHORSZ] S. Argryros, D. Freeman, R. Haydon, E. Odell, Th. Raikoftsalis, Th. Schlumprecht and D. Z. Zisimopoulou, Embedding Banach spaces into spaces with very few operators, J. Functional Anal. 262 (2012), no. 3, 825-849. [AH] S. A. Argyros and R. G. Haydon, A hereditarily indecomposable L∞ -space that solves the scalar-plus-compact problem, Acta Math. 206 (2011), 1-54. [BD] J. Bourgain and F. Delbaen, A class of special L∞ spaces, Acta Math. 145 (1980), 155-176. [FOS] D. Freeman, E. Odell and Th. Schlumprecht, The universality of ℓ1 as a dual space, Math. Annalen. 351 (2011), no. 1, 149-186. [GL] G. Godefroy and D. Li, Some natural families of M -ideals, Math. Scand. 66 (1990), 249-263. [JRZ] W. B. Johnson, H. P. Rosenthal and M. Zippin, On bases, finite dimensional decompositions and weaker structures in Banach spaces, Israel J. Math. 9 (1971), 488-506. [LS] D. Lewis and C. Stegall, Banach spaces whose duals are isomorphic to ℓ1 (Γ), J. Functional Analysis 12 (1973), 177-187. [LP] J. Lindenstrauss and A. Pelczy´ nski, Absolutely summing operators in Lp -spaces and their applications, Studia Math. 29 (1968), 275-326. [LR] J. Lindenstrauss and H. P. Rosenthal, The Lp spaces, Israel J. Math. 7 (1969), 325-349. [P] A. Pelczyn´ski, On Banach spaces containing L1 (µ), Studia Math. 30 (1968), 231-246. [S] C. Stegall, Banach spaces whose duals contain ℓ1 (Γ) with applications to the study of dual L1 (µ) spaces, Trans. Amer. Math. Soc. 176 (1973), 463-477. [T] B. S. Tsirelson, Not every Banach space contains ℓp or c0 , Functional Anal. Appl. 8 (1974), 138-141.1 National Technical University of Athens, Faculty of Applied Sciences, Department of Mathematics, Zografou Campus, 157 80, Athens, Greece E-mail address: [email protected] E-mail address: [email protected] E-mail address: [email protected]